Abstract
This paper applies Dehornoy et al.’s Z theorem and its variant, called the compositional Z theorem, to prove confluence of Parigot’s \(\lambda \mu \)-calculi extended by the simplification rules. First, it is proved that Baba et al.’s modified complete developments for the call-by-name and the call-by-value variants of the \(\lambda \mu \)-calculus with the renaming rule, which is one of the simplification rules, satisfy the Z property. It gives new confluence proofs for them by the Z theorem. Secondly, it is shown that the compositional Z theorem can be applied to prove confluence of the call-by-name and the call-by-value \(\lambda \mu \)-calculi with both simplification rules, the renaming and the \(\mu \eta \)-rules, whereas it is hard to apply the ordinary parallel reduction technique or the original Z theorem by one-pass definition of mappings for these variants.
Similar content being viewed by others
References
Baba, K., S. Hirokawa, and K. Fujita, Parallel reduction in type free \(\lambda \mu \)-calculus, Electronic Notes in Theoretical Computer Science 42:52–66, 2001.
Dehornoy, P., and V. van Oostrom, Z. Proving confluence by monotonic single-step upperbound functions, in Logical Models of Reasoning and Computation (LMRC-08), 2008.
Fujita, K., Explicitly typed \(\lambda \mu \)-calculus for polymorphism and call-by-value, in Typed Lambda Calculi and Applications, 4th International Conference (TLCA 1999), volume 1581 of Lecture Notes in Computer Science, 1999, pp. 162–176.
Fujita, K., Domain-free \(\lambda \mu \)-calculus, Theoretical Informatics and Applications 34:433–466, 2000.
Nakazawa, K., Confluency and strong normalizability of call-by-value \(\lambda \mu \)-calculus, Theoretical Computer Science 290:429–463, 2003.
Nakazawa, K., and K. Fujita, Compositional Z: Confluence proofs for permutative conversion, Studia Logica 104:1205–1224, 2016.
Ong, C.-H.L., and C. A. Stewart, A curry-howard foundation for functional computation with control, in 24th Annual ACM Symposium of Principles of Programming Languages (POPL 1997), 1997, pp. 215–227.
Parigot, M., \(\lambda \mu \)-calculus: an algorithmic interpretation of classical natural deduction, in Proceedings of the International Conference on Logic Programming and Automated Reasoning (LPAR ’92), volume 624 of Lecture Notes in Computer Science, Springer, 1992, pp. 190–201.
Saurin, A., Typing streams in the \(\Lambda \mu \)-calculus, ACM Transactions on Computational Logic 11:1–34, 2010.
Acknowledgements
This work was partially supported by Grants-in-Aid for Scientific Research KAKENHI 18K11161 (to Yuki Honda and Koji Nakazawa) and 17K05343 and 20K03711 (to Ken-etsu Fujita).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Daniele Mundici
Rights and permissions
About this article
Cite this article
Honda, Y., Nakazawa, K. & Fujita, Ke. Confluence Proofs of Lambda-Mu-Calculi by Z Theorem. Stud Logica 109, 917–936 (2021). https://doi.org/10.1007/s11225-020-09931-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-020-09931-0